91Ó°ÊÓ

In Exercises \(11-24,\) find all solutions of each equation. $$\sin x=\frac{\sqrt{3}}{2}$$

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$\cos ^{2} 75^{\circ}-\sin ^{2} 75^{\circ}$$

In Exercises \(11-24,\) find all solutions of each equation. $$5 \sin \theta+1=3 \sin \theta$$

Verify each identity. $$(\sin \theta-\cos \theta)^{2}=1-\sin 2 \theta$$

Verify identity \(\frac{\sin x-\sin y}{\sin x+\sin y}=\tan \frac{x-y}{2} \cot \frac{x+y}{2}\)

Verify each identity. $$\sin 4 t=4 \sin t \cos ^{3} t-4 \sin ^{3} t \cos t$$

Find the exact value of the following under the given conditions: \(a, \cos (\alpha+\beta)\) \(b . \sin (\alpha+\beta)\) \(c . \tan (\alpha+\beta)\) \(\tan \alpha=-\frac{4}{3}, \alpha\) lies in quadrant II, and \(\cos \beta=\frac{2}{3}, \beta\) lies in quadrant I.

Rewrite each expression as a simplified expression containing one term. $$\sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta$$

Rewrite each expression as a simplified expression containing one term. $$\frac{\cos (\alpha-\beta)+\cos (\alpha+\beta)}{-\sin (\alpha-\beta)+\sin (\alpha+\beta)}$$

Rewrite each expression as a simplified expression containing one term. \(\sin \left(\frac{\pi}{3}-\alpha\right) \cos \left(\frac{\pi}{3}+\alpha\right)+\cos \left(\frac{\pi}{3}-\alpha\right) \sin \left(\frac{\pi}{3}+\alpha\right)\) (Do not use four different identities to solve this exercise.)